Question

Show that [x.y] is $\Omega \left(x,y\right)$

Short answer

$\left[x,y\right]is\Omega \left(x,y\right)$Hence,

Step-by-step solution

Step 1:

Let $f(x,y)=\left[xy\right]$and

Let data-custom-editor="chemistry" $k_1=k_2=1$

We know $x>k_1$and$y>k_2$

$\Rightarrow x>1$and y > 1

Step 2:

Applying Big-Omega notation definition we get

$\left|f\right(x,y\left)\right|=\left[xy\right]$

We know that$\left|a\right|\ge a$ for any value of a {Property of ceiling function}

$$\begin{gathered} \Rightarrow \left|f\right(x,y\left)\right|\ge xy \\ \Rightarrow \left|f\right(x,y\left)\right|\ge \left|xy\right| \end{gathered}$$

Comparing the inequality with$\left|f\right(x,y\left)\right|\ge C\left|g\right(x,y\left)\right|$ we get and

$\left[xy\right]is\Omega \left(xy\right)$